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Theorem opsqrlem6 29132
Description: Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
opsqrlem2.1 𝑇 ∈ HrmOp
opsqrlem2.2 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))
opsqrlem2.3 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
opsqrlem6.4 𝑇op Iop
Assertion
Ref Expression
opsqrlem6 (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
Distinct variable group:   𝑥,𝑦,𝑇
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem opsqrlem6
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . 3 (𝑗 = 1 → (𝐹𝑗) = (𝐹‘1))
21breq1d 4695 . 2 (𝑗 = 1 → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘1) ≤op Iop ))
3 fveq2 6229 . . 3 (𝑗 = (𝑘 + 1) → (𝐹𝑗) = (𝐹‘(𝑘 + 1)))
43breq1d 4695 . 2 (𝑗 = (𝑘 + 1) → ((𝐹𝑗) ≤op Iop ↔ (𝐹‘(𝑘 + 1)) ≤op Iop ))
5 fveq2 6229 . . 3 (𝑗 = 𝑁 → (𝐹𝑗) = (𝐹𝑁))
65breq1d 4695 . 2 (𝑗 = 𝑁 → ((𝐹𝑗) ≤op Iop ↔ (𝐹𝑁) ≤op Iop ))
7 opsqrlem2.1 . . . 4 𝑇 ∈ HrmOp
8 opsqrlem2.2 . . . 4 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))
9 opsqrlem2.3 . . . 4 𝐹 = seq1(𝑆, (ℕ × { 0hop }))
107, 8, 9opsqrlem2 29128 . . 3 (𝐹‘1) = 0hop
11 idleop 29118 . . 3 0hopop Iop
1210, 11eqbrtri 4706 . 2 (𝐹‘1) ≤op Iop
13 idhmop 28969 . . . . . . . 8 Iop ∈ HrmOp
147, 8, 9opsqrlem4 29130 . . . . . . . . 9 𝐹:ℕ⟶HrmOp
1514ffvelrni 6398 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ HrmOp)
16 hmopd 29009 . . . . . . . 8 (( Iop ∈ HrmOp ∧ (𝐹𝑘) ∈ HrmOp) → ( Iopop (𝐹𝑘)) ∈ HrmOp)
1713, 15, 16sylancr 696 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)) ∈ HrmOp)
18 eqid 2651 . . . . . . . 8 (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))
19 hmopco 29010 . . . . . . . 8 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp ∧ (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2018, 19mp3an3 1453 . . . . . . 7 ((( Iopop (𝐹𝑘)) ∈ HrmOp ∧ ( Iopop (𝐹𝑘)) ∈ HrmOp) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
2117, 17, 20syl2anc 694 . . . . . 6 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp)
22 leopsq 29116 . . . . . . 7 (( Iopop (𝐹𝑘)) ∈ HrmOp → 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))))
2317, 22syl 17 . . . . . 6 (𝑘 ∈ ℕ → 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))))
24 opsqrlem6.4 . . . . . . . 8 𝑇op Iop
25 leop3 29112 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ Iop ∈ HrmOp) → (𝑇op Iop ↔ 0hopop ( Iopop 𝑇)))
267, 13, 25mp2an 708 . . . . . . . 8 (𝑇op Iop ↔ 0hopop ( Iopop 𝑇))
2724, 26mpbi 220 . . . . . . 7 0hopop ( Iopop 𝑇)
28 hmopd 29009 . . . . . . . . 9 (( Iop ∈ HrmOp ∧ 𝑇 ∈ HrmOp) → ( Iopop 𝑇) ∈ HrmOp)
2913, 7, 28mp2an 708 . . . . . . . 8 ( Iopop 𝑇) ∈ HrmOp
30 leopadd 29119 . . . . . . . 8 ((((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ ( Iopop 𝑇) ∈ HrmOp) ∧ ( 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∧ 0hopop ( Iopop 𝑇))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3129, 30mpanl2 717 . . . . . . 7 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ ( 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∧ 0hopop ( Iopop 𝑇))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3227, 31mpanr2 720 . . . . . 6 (((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) ∈ HrmOp ∧ 0hopop (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘)))) → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
3321, 23, 32syl2anc 694 . . . . 5 (𝑘 ∈ ℕ → 0hopop ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)))
34 2cn 11129 . . . . . . . . . 10 2 ∈ ℂ
35 hmopf 28861 . . . . . . . . . . 11 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘): ℋ⟶ ℋ)
3615, 35syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (𝐹𝑘): ℋ⟶ ℋ)
37 homulcl 28746 . . . . . . . . . 10 ((2 ∈ ℂ ∧ (𝐹𝑘): ℋ⟶ ℋ) → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
3834, 36, 37sylancr 696 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)): ℋ⟶ ℋ)
39 hmopf 28861 . . . . . . . . . . 11 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
407, 39ax-mp 5 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
41 fco 6096 . . . . . . . . . . 11 (((𝐹𝑘): ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
4236, 36, 41syl2anc 694 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)
43 hosubcl 28760 . . . . . . . . . 10 ((𝑇: ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
4440, 42, 43sylancr 696 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
45 hmopf 28861 . . . . . . . . . . . 12 ( Iop ∈ HrmOp → Iop : ℋ⟶ ℋ)
4613, 45ax-mp 5 . . . . . . . . . . 11 Iop : ℋ⟶ ℋ
47 homulcl 28746 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ) → (2 ·op Iop ): ℋ⟶ ℋ)
4834, 46, 47mp2an 708 . . . . . . . . . 10 (2 ·op Iop ): ℋ⟶ ℋ
49 hosubsub4 28805 . . . . . . . . . 10 (((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
5048, 49mp3an1 1451 . . . . . . . . 9 (((2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
5138, 44, 50syl2anc 694 . . . . . . . 8 (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
52 hosubcl 28760 . . . . . . . . . . . . . . 15 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
5342, 38, 52syl2anc 694 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ)
54 hoadd32 28770 . . . . . . . . . . . . . . 15 (( Iop : ℋ⟶ ℋ ∧ (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ) → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
5546, 46, 54mp3an13 1455 . . . . . . . . . . . . . 14 ((((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
5653, 55syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
57 ho2times 28806 . . . . . . . . . . . . . . 15 ( Iop : ℋ⟶ ℋ → (2 ·op Iop ) = ( Iop +op Iop ))
5846, 57ax-mp 5 . . . . . . . . . . . . . 14 (2 ·op Iop ) = ( Iop +op Iop )
5958oveq1i 6700 . . . . . . . . . . . . 13 ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) = (( Iop +op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))))
6056, 59syl6eqr 2703 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
61 hoaddsubass 28802 . . . . . . . . . . . . . 14 (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6248, 61mp3an1 1451 . . . . . . . . . . . . 13 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6342, 38, 62syl2anc 694 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ((2 ·op Iop ) +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
6460, 63eqtr4d 2688 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))))
6564oveq1d 6705 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇))
66 hoaddcl 28745 . . . . . . . . . . . 12 (( Iop : ℋ⟶ ℋ ∧ (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘))): ℋ⟶ ℋ) → ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ)
6746, 53, 66sylancr 696 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ)
68 hoaddsubass 28802 . . . . . . . . . . . 12 ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ ∧ Iop : ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
6946, 40, 68mp3an23 1456 . . . . . . . . . . 11 (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))): ℋ⟶ ℋ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
7067, 69syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op Iop ) −op 𝑇) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
71 hoaddcl 28745 . . . . . . . . . . . 12 (((2 ·op Iop ): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
7248, 42, 71sylancr 696 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ)
73 hosubsub4 28805 . . . . . . . . . . . 12 ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7440, 73mp3an3 1453 . . . . . . . . . . 11 ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7572, 38, 74syl2anc 694 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) −op 𝑇) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7665, 70, 753eqtr3d 2693 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
77 hosubadd4 28801 . . . . . . . . . . . 12 ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7840, 77mpanr1 719 . . . . . . . . . . 11 ((((2 ·op Iop ): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
7948, 78mpanl1 716 . . . . . . . . . 10 (((2 ·op (𝐹𝑘)): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ) → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
8038, 42, 79syl2anc 694 . . . . . . . . 9 (𝑘 ∈ ℕ → (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (((2 ·op Iop ) +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((2 ·op (𝐹𝑘)) +op 𝑇)))
8176, 80eqtr4d 2688 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (((2 ·op Iop ) −op (2 ·op (𝐹𝑘))) −op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
82 halfcn 11285 . . . . . . . . . . . 12 (1 / 2) ∈ ℂ
83 homulcl 28746 . . . . . . . . . . . 12 (((1 / 2) ∈ ℂ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
8482, 44, 83sylancr 696 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ)
85 hoadddi 28790 . . . . . . . . . . . 12 ((2 ∈ ℂ ∧ (𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
8634, 85mp3an1 1451 . . . . . . . . . . 11 (((𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
8736, 84, 86syl2anc 694 . . . . . . . . . 10 (𝑘 ∈ ℕ → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
88 2ne0 11151 . . . . . . . . . . . . . 14 2 ≠ 0
8934, 88recidi 10794 . . . . . . . . . . . . 13 (2 · (1 / 2)) = 1
9089oveq1i 6700 . . . . . . . . . . . 12 ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
91 homulass 28789 . . . . . . . . . . . . . 14 ((2 ∈ ℂ ∧ (1 / 2) ∈ ℂ ∧ (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ) → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
9234, 82, 91mp3an12 1454 . . . . . . . . . . . . 13 ((𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
9344, 92syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((2 · (1 / 2)) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
94 homulid2 28787 . . . . . . . . . . . . 13 ((𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))): ℋ⟶ ℋ → (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9544, 94syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9690, 93, 953eqtr3a 2709 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))) = (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))
9796oveq2d 6706 . . . . . . . . . 10 (𝑘 ∈ ℕ → ((2 ·op (𝐹𝑘)) +op (2 ·op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9887, 97eqtrd 2685 . . . . . . . . 9 (𝑘 ∈ ℕ → (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))) = ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))
9998oveq2d 6706 . . . . . . . 8 (𝑘 ∈ ℕ → ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op ((2 ·op (𝐹𝑘)) +op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
10051, 81, 993eqtr4d 2695 . . . . . . 7 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
101 hoaddcl 28745 . . . . . . . . 9 (((𝐹𝑘): ℋ⟶ ℋ ∧ ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))): ℋ⟶ ℋ) → ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ)
10236, 84, 101syl2anc 694 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ)
103 hosubdi 28795 . . . . . . . . 9 ((2 ∈ ℂ ∧ Iop : ℋ⟶ ℋ ∧ ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ) → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
10434, 46, 103mp3an12 1454 . . . . . . . 8 (((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))): ℋ⟶ ℋ → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
105102, 104syl 17 . . . . . . 7 (𝑘 ∈ ℕ → (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))) = ((2 ·op Iop ) −op (2 ·op ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
106100, 105eqtr4d 2688 . . . . . 6 (𝑘 ∈ ℕ → (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)) = (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
107 hosubcl 28760 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
10846, 36, 107sylancr 696 . . . . . . . . 9 (𝑘 ∈ ℕ → ( Iopop (𝐹𝑘)): ℋ⟶ ℋ)
109 hocsubdir 28772 . . . . . . . . . 10 (( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11046, 109mp3an1 1451 . . . . . . . . 9 (((𝐹𝑘): ℋ⟶ ℋ ∧ ( Iopop (𝐹𝑘)): ℋ⟶ ℋ) → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
11136, 108, 110syl2anc 694 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))))
112 hmoplin 28929 . . . . . . . . . . . . . . 15 ( Iop ∈ HrmOp → Iop ∈ LinOp)
11313, 112ax-mp 5 . . . . . . . . . . . . . 14 Iop ∈ LinOp
114 hoddi 28977 . . . . . . . . . . . . . 14 (( Iop ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
115113, 46, 114mp3an12 1454 . . . . . . . . . . . . 13 ((𝐹𝑘): ℋ⟶ ℋ → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
11636, 115syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ( Iop ∘ ( Iopop (𝐹𝑘))) = (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))))
11746hoid1i 28776 . . . . . . . . . . . . . 14 ( Iop ∘ Iop ) = Iop
118117a1i 11 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ Iop ) = Iop )
119 hoico2 28744 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
12036, 119syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ( Iop ∘ (𝐹𝑘)) = (𝐹𝑘))
121118, 120oveq12d 6708 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (( Iop ∘ Iop ) −op ( Iop ∘ (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
122116, 121eqtrd 2685 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop ∘ ( Iopop (𝐹𝑘))) = ( Iopop (𝐹𝑘)))
123 hmoplin 28929 . . . . . . . . . . . . . 14 ((𝐹𝑘) ∈ HrmOp → (𝐹𝑘) ∈ LinOp)
12415, 123syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) ∈ LinOp)
125 hoddi 28977 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ LinOp ∧ Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
12646, 125mp3an2 1452 . . . . . . . . . . . . 13 (((𝐹𝑘) ∈ LinOp ∧ (𝐹𝑘): ℋ⟶ ℋ) → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
127124, 36, 126syl2anc 694 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
128 hoico1 28743 . . . . . . . . . . . . . 14 ((𝐹𝑘): ℋ⟶ ℋ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
12936, 128syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ Iop ) = (𝐹𝑘))
130129oveq1d 6705 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (((𝐹𝑘) ∘ Iop ) −op ((𝐹𝑘) ∘ (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
131127, 130eqtrd 2685 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘))) = ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘))))
132122, 131oveq12d 6708 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))))
13336, 46jctil 559 . . . . . . . . . . 11 (𝑘 ∈ ℕ → ( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ))
134 hosubadd4 28801 . . . . . . . . . . 11 ((( Iop : ℋ⟶ ℋ ∧ (𝐹𝑘): ℋ⟶ ℋ) ∧ ((𝐹𝑘): ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ)) → (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
135133, 36, 42, 134syl12anc 1364 . . . . . . . . . 10 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) −op ((𝐹𝑘) −op ((𝐹𝑘) ∘ (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
136132, 135eqtrd 2685 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
137 ho2times 28806 . . . . . . . . . . 11 ((𝐹𝑘): ℋ⟶ ℋ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
13836, 137syl 17 . . . . . . . . . 10 (𝑘 ∈ ℕ → (2 ·op (𝐹𝑘)) = ((𝐹𝑘) +op (𝐹𝑘)))
139138oveq2d 6706 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op ((𝐹𝑘) +op (𝐹𝑘))))
140 hoaddsubass 28802 . . . . . . . . . . 11 (( Iop : ℋ⟶ ℋ ∧ ((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
14146, 140mp3an1 1451 . . . . . . . . . 10 ((((𝐹𝑘) ∘ (𝐹𝑘)): ℋ⟶ ℋ ∧ (2 ·op (𝐹𝑘)): ℋ⟶ ℋ) → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
14242, 38, 141syl2anc 694 . . . . . . . . 9 (𝑘 ∈ ℕ → (( Iop +op ((𝐹𝑘) ∘ (𝐹𝑘))) −op (2 ·op (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
143136, 139, 1423eqtr2d 2691 . . . . . . . 8 (𝑘 ∈ ℕ → (( Iop ∘ ( Iopop (𝐹𝑘))) −op ((𝐹𝑘) ∘ ( Iopop (𝐹𝑘)))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
144111, 143eqtrd 2685 . . . . . . 7 (𝑘 ∈ ℕ → (( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) = ( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))))
145144oveq1d 6705 . . . . . 6 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (( Iop +op (((𝐹𝑘) ∘ (𝐹𝑘)) −op (2 ·op (𝐹𝑘)))) +op ( Iopop 𝑇)))
1467, 8, 9opsqrlem5 29131 . . . . . . . 8 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) = ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))
147146oveq2d 6706 . . . . . . 7 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) = ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘)))))))
148147oveq2d 6706 . . . . . 6 (𝑘 ∈ ℕ → (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))) = (2 ·op ( Iopop ((𝐹𝑘) +op ((1 / 2) ·op (𝑇op ((𝐹𝑘) ∘ (𝐹𝑘))))))))
149106, 145, 1483eqtr4d 2695 . . . . 5 (𝑘 ∈ ℕ → ((( Iopop (𝐹𝑘)) ∘ ( Iopop (𝐹𝑘))) +op ( Iopop 𝑇)) = (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
15033, 149breqtrd 4711 . . . 4 (𝑘 ∈ ℕ → 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1)))))
151 peano2nn 11070 . . . . . . 7 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
15214ffvelrni 6398 . . . . . . 7 ((𝑘 + 1) ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
153151, 152syl 17 . . . . . 6 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ∈ HrmOp)
154 hmopd 29009 . . . . . 6 (( Iop ∈ HrmOp ∧ (𝐹‘(𝑘 + 1)) ∈ HrmOp) → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
15513, 153, 154sylancr 696 . . . . 5 (𝑘 ∈ ℕ → ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp)
156 2re 11128 . . . . . 6 2 ∈ ℝ
157 2pos 11150 . . . . . 6 0 < 2
158 leopmul 29121 . . . . . 6 ((2 ∈ ℝ ∧ ( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp ∧ 0 < 2) → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
159156, 157, 158mp3an13 1455 . . . . 5 (( Iopop (𝐹‘(𝑘 + 1))) ∈ HrmOp → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
160155, 159syl 17 . . . 4 (𝑘 ∈ ℕ → ( 0hopop ( Iopop (𝐹‘(𝑘 + 1))) ↔ 0hopop (2 ·op ( Iopop (𝐹‘(𝑘 + 1))))))
161150, 160mpbird 247 . . 3 (𝑘 ∈ ℕ → 0hopop ( Iopop (𝐹‘(𝑘 + 1))))
162 leop3 29112 . . . 4 (((𝐹‘(𝑘 + 1)) ∈ HrmOp ∧ Iop ∈ HrmOp) → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
163153, 13, 162sylancl 695 . . 3 (𝑘 ∈ ℕ → ((𝐹‘(𝑘 + 1)) ≤op Iop ↔ 0hopop ( Iopop (𝐹‘(𝑘 + 1)))))
164161, 163mpbird 247 . 2 (𝑘 ∈ ℕ → (𝐹‘(𝑘 + 1)) ≤op Iop )
1652, 4, 6, 12, 164nn1suc 11079 1 (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {csn 4210   class class class wbr 4685   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   / cdiv 10722  cn 11058  2c2 11108  seqcseq 12841  chil 27904   +op chos 27923   ·op chot 27924  op chod 27925   0hop ch0o 27928   Iop chio 27929  LinOpclo 27932  HrmOpcho 27935  op cleo 27943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054  ax-hilex 27984  ax-hfvadd 27985  ax-hvcom 27986  ax-hvass 27987  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991  ax-hvmulass 27992  ax-hvdistr1 27993  ax-hvdistr2 27994  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his2 28068  ax-his3 28069  ax-his4 28070  ax-hcompl 28187
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-cn 21079  df-cnp 21080  df-lm 21081  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cfil 23099  df-cau 23100  df-cmet 23101  df-grpo 27475  df-gid 27476  df-ginv 27477  df-gdiv 27478  df-ablo 27527  df-vc 27542  df-nv 27575  df-va 27578  df-ba 27579  df-sm 27580  df-0v 27581  df-vs 27582  df-nmcv 27583  df-ims 27584  df-dip 27684  df-ssp 27705  df-ph 27796  df-cbn 27847  df-hnorm 27953  df-hba 27954  df-hvsub 27956  df-hlim 27957  df-hcau 27958  df-sh 28192  df-ch 28206  df-oc 28237  df-ch0 28238  df-shs 28295  df-pjh 28382  df-hosum 28717  df-homul 28718  df-hodif 28719  df-h0op 28735  df-iop 28736  df-lnop 28828  df-hmop 28831  df-leop 28839
This theorem is referenced by: (None)
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